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Coordinate geometry
Points, lines and distances on a grid. It sounds like a lot of formulas, but they all come from two ideas: slope is rise over run, and distance is the Pythagorean theorem drawn on graph paper.
Slope is rise over run
slope = (y₂ − y₁) ÷ (x₂ − x₁)
A point is written (x, y): first how far across, then how far up. The slope of a line says how steep it is: how much it goes up (rise) for every step to the right (run). Positive slopes go up to the right; negative slopes go down.
y on top, x on the bottom
Rise is the change in y; run is the change in x. Subtract in the same order on top and bottom (second point minus first point), and the sign takes care of itself.
How to solve them
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1
Slope from two points
Subtract the y-values, subtract the x-values in the same order, and divide.
Next step: y = mx + b- (1, 3) and (5, 11)
- m = (11 − 3) ÷ (5 − 1)
- m = 8 ÷ 4
- m = 2
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2
y = mx + b: the slope and where the line crosses
In y = mx + b, m is the slope and b is the y-intercept, where the line crosses the y-axis (when x = 0). To find a point on the line, put an x-value in and work out y.
Next step: Line from pointsRead ity = 3x − 4
slope 3, crosses at −4
Find a yy = 3x − 4 when x = 2
3 × 2 − 4 = 2
Flat or steepy = 5 and y = −2x
slope 0 (flat) · slope −2 (down)
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3
The equation of a line from two points
Find the slope first. Then put one point into y = mx + b to find b.
Next step: MidpointExampleFind the line through (0, 5) and (2, 1).
- Slope: (1 − 5) ÷ (2 − 0) = −4 ÷ 2 = −2.
- At x = 0 the line is at 5, so b = 5.
- y = −2x + 5. Check with (2, 1): −4 + 5 = 1.
Answer y = −2x + 5
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4
Midpoint: average the x’s, average the y’s
The point halfway between two points has the average of their x-values and the average of their y-values.
((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
Next step: DistanceExampleWhat is the midpoint of (−3, 4) and (7, 10)?
- x: (−3 + 7) ÷ 2 = 4 ÷ 2 = 2.
- y: (4 + 10) ÷ 2 = 14 ÷ 2 = 7.
Answer (2, 7)
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5
Distance: the Pythagorean theorem on the grid
The horizontal and vertical gaps between two points are the legs of a right triangle; the distance is its hypotenuse.
Next step: Parallel & perpendicular- (1, 2) to (7, 10)
- legs: 7 − 1 = 6 and 10 − 2 = 8
- distance = √(36 + 64)
- √100 = 10
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6
Parallel and perpendicular lines
Parallel lines have the same slope. Perpendicular lines meet at a right angle, and their slopes are negative reciprocals: flip the fraction and change the sign.
Parallel to y = 4x + 1Same slope
4
Perpendicular to y = 4x + 1Flip and change sign
−1/4
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Run over rise
Slope from (1, 3) to (5, 11):
Wrong4 ÷ 8 = 0.5
Right8 ÷ 4 = 2
Subtracting in different orders
Slope from (2, −1) to (6, 7):
Wrong(7 − (−1)) ÷ (2 − 6) = −2
Right(7 − (−1)) ÷ (6 − 2) = 2
Midpoint as the difference
Midpoint of (2, 4) and (8, 10):
Wrong(6, 6)
Right(5, 7)
Distance as the sum of the gaps
From (0, 0) to (3, 4):
Wrong3 + 4 = 7
Right√(9 + 16) = 5
5 questions at test level
Pick an answer, then read why. No timer, no account.
What is the slope of the line through (2, 3) and (6, 15)?
Show the answer
B. Rise: 15 − 3 = 12. Run: 6 − 2 = 4. Slope = 12 ÷ 4 = 3. 1/3 divides run by rise, and 12 forgets to divide. The slope is 3.
What is the y-intercept of the line y = 4x − 9?
Show the answer
A. In y = mx + b the y-intercept is b, the number without x: −9 (where x = 0, y = −9). 4 is the slope. The y-intercept is −9.
What is the midpoint of (−4, 6) and (8, 2)?
Show the answer
B. Average the x-values: (−4 + 8) ÷ 2 = 2. Average the y-values: (6 + 2) ÷ 2 = 4. (6, −2) is half the difference, and (12, −4) is the difference. The midpoint is (2, 4).
What is the distance between (1, 1) and (10, 13)?
Show the answer
C. The gaps are 10 − 1 = 9 and 13 − 1 = 12. Distance = √(81 + 144) = √225 = 15 (the 9-12-15 triple). 21 adds the gaps. The distance is 15.
Which equation describes the line through (0, −2) with a slope of 5?
Show the answer
C. The slope is m = 5, and the line crosses the y-axis at (0, −2), so b = −2: y = 5x − 2. y = −2x + 5 swaps the slope and the intercept. The equation is y = 5x − 2.
10 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 10 cards known.
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